In this task, students will observe a small square inscribed in a larger square. Students will use their prior knowledge of areas of squares, rectangles, and triangles to determine the area of the inscribed square.

The purpose of this task is for students to apply their understanding and knowledge of determining areas of composite figures and congruence in a context that will allow them to bump into the Pythagorean relationship. Students will build fluency in determining areas of squares and triangles while recalling or realizing that congruent shapes occupy the same area before and after translations and rotations. Students will reflect and use the inverse relationship between the area of a square and the side length of the square.

Some ideas that may emerge through this task include:

• Area of composite figures;
• The Pythagorean relationship; and,
• Congruence.
• Translations / rotations.
• Squaring numbers and their square roots.
• The sum of the area of the squares constructed using the short legs of a right triangle is equivalent to the area of the square constructed using the longest leg.

Show students the following video:

Then, ask students:

What do you notice?

What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Replaying the video and/or leaving a screenshot from the video up can be helpful here.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other means to ensure students know that their voice is acknowledged and appreciated.

Some of the noticing and wondering that may come up includes:

• There are two squares.
• Some line segments are equal and some are not.
• I see triangles and squares.
• I see a diamond.
• I wonder if those triangles are the same?
• I wonder if how much area is the big square?
• I wonder how much area is the smaller square?

After we have heard students and demonstrated that we value their voice, we can land on the first question we will challenge them with:

Show students the image below.

How many of the small squares will fit in the large square?

We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to estimate dimensions and use their knowledge of area.

Encourage students to share their estimates, however avoid sharing their justification just yet. We do not want to rob other students of their thinking.

Prompt students by asking:

If you were to have more information to help narrow your prediction/make your prediction more accurate what extra information would you want?

Engage your class in Think-Pair-Share where they Think independently on what they would like to have to update their prediction, Pair and share their thinking with a partner, finally share their thinking with the class.

Show the More Information images one at time pausing and prompting your students to share their strategies.

Image #1

Image #2

With this information and confirming that the largest square is in fact a square, ask students to update their prediction while showing their mathematical thinking.

You might also consider sharing this Mathigon Polypad virtual manipulative template so that students can interact with the figure as they update their original prediction.

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Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

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Consider sharing the following reveal video with your students:
Keep in mind this is the same video as the making connections video above.

Consider leaving the following screenshot of the final frame up for students to reflect on.

Students will complete the following consolidation prompts independently or in small groups.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

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